Dette er ganske interessant. Hvis jeg fortalte dig, at fra en selvmodsigelse følger alting, ville du så være umiddelbart enig? Nej vel. Det var jeg heller ikke, men så mødte jeg dette bevis.
assumption
from (1) by conjunction elimination
from (1) by conjunction elimination
from (2) by disjunction introduction
from (3) and (4) by disjunctive syllogism
from (5) by conditional proof (discharging assumption 1)
Dante forklarer det:
1. (p & ~p) [Assumption]
2. p [1, conjunctive elimination]
3. ~p [1, conjunctive elimination]
4. p v q [2, disjunctive introduction]
5. q [3, 4, disjunctive syllogism]
6. (p & ~p) -> q [5, conditional proof](1): you assume a contradiction.
(2) & (3): Now, the thing about any conjunction whatsoever is you can separate the conjuncts as true propositions. If (p & q) is true, then that means that both p is true and q is true, and you can separately assert truly that p and q. This follows obviously because a conjunction is true only if and only if both conjuncts is true.
(4): For any true proposition p, it is also true that p v q. The thing about a disjunction is that at least one disjunct is true. So, if you have a true proposition, then plainly enough, any disjunction involving it is true. Hence, you can introduce any true disjunction from a true proposition.
(5): q follows from disjunctive syllogism. A disjunctive syllogism is of the form “Either p or q, not p, therefore q.” An example: “Either the cat is on the mat or on the floor, it is not on the mat, therefore, it is on the floor.” Now, the disjunction in question is true and per (3), p is false. Hence, by disjunctive syllogism, it follows that q is true.
(6): This argument applies for any proposition p and q; hence, it shows a proof that a contradiction entails any proposition whatsoever.
Sejt ikke?